I think that they are pretty straightforward to assemble without any
instructions  just cut them out carefully, fold along the black
lines, and tape. In some cases you'll need to print multiple copies
of a template. The parenthesized numbers before each name tells you
what each vertex should look like: in a (3,4,3,4)
cuboctahedron, for instance, each vertex should be surrounded by a
triangle, a square, another triangle, and other square, in order as
you go around the vertex. Knowing this can be helpful when you're
trying to fold the pieces up and figure out what pairs of edges match
up.

For best results, print these templates out on cardstock, or print
them on regular paper and use a pin to transfer the template to
something heavier. Each template is fitted within the printable area
of a Tektronix Phaser 360 color printer (which is what I used in
making mine). If your printer has a smaller printable area, some
edges may get cut off. Sorry.

All of the models have a common edge length, so relative sizes are
meaningful. The squares on the smallest (the cuboctahedron) are the
same size as the squares of the largest (the truncated
icosidodecahedron). This means that some of the polyhedra turn out
very large, and take several sheets of paper. The largest one
requires seven pages of template and is roughly nine inches in
diameter when completed.

Start by making the Platonic solids and the smaller Archimedeans
first. The larger ones are trickier to cut out and assemble.

(3,8,8) truncated cube

(3,4,4,4) rhombicuboctahedron

(4,6,8) truncated cuboctahedron

Print two copies of this page to get four pieces, two of which are
individual squares. The bottom side of the bottom hexagon of one big
piece fits up against the free side of the top square of the other
piece, leaving a square hole on each side.

(3,3,3,3,4) snub cuboctahedron

This shape comes from just a single piece. You can imagine creating
this shape by taking the (3,4,4,4) rhombicuboctahedron and turning
each yellow square into two yellow triangles. That's why there are
two different colors of triangle in this model  the blue ones are
the eight blue triangles already present in the rhombicuboctahedron.

(3,5,3,5) icosidodecahedron

(5,6,6) truncated icosahedron

Everyone's favorite polyhedron  the shape of soccerballs and
buckyballs.

You need two copies of this page. Notice how each big piece consists
of a central pentagon with double-hexagon "arms" radiating out, except
that one of the arms has only a single hex. This is where the
individual hexagon should be attached. Once this is done you'll have
two star-shaped pieces that fold up into big bowls, and then the rims
of the bowls can be matched up.

The most difficult part of making this one is cutting it out, because
of those lightning-bolt-shaped "alleyways" that come in from the
perimeter.

(3,4,5,4) rhombicosidodecahedron

Print out two copies to get the six total pieces you'll need. The two
star shapes fold up into shallow bowls. Two copies of the snake plus
the two individual squares can be strung together to make a belt
around the equator that joins the bowls together  just continue the
alternating triangles and squares pattern.

(4,6,10) truncated icosidodecahedron

The largest shape, and the only one to require two different pages.
You'll need two copes of shape A on the left, which will make
two very shallow bowls, or caps. Five copies of shape B,
on the right, will join together in a rotationally symmetric way to
make a very wide band that joins the two caps together.

(3,3,3,3,5) snub icosidodecahedron

Print two copies to make four identical pieces. This is probably the
most difficult to assemble because the faces are small and the
dihedral angles are large.

Each of the four pieces has threefold rotational symmetry (once it's
taped up into a shallow bowl), except for a single blue triangle that
sticks out in one place. This is also the case for the (3,10,10)
truncated dodecahedron; the four pieces fit together in analogous
ways. You should be able to fit two pieces together to make a kind of
U-shape; a second identical U-shape will complete the polyhedron with
a 90-degree rotation (think of how a baseball is stitched together
from two pieces). Keep in mind that a blue triangle should be
adjacent to three yellow triangles, and that each yellow triangle
should be adjacent to exactly one blue triangle, one yellow triangle,
and one purple pentagon.